16 1. Various Ways of Representing Surfaces and Examples

Figure 1.12. From the sphere to a disc via geographic coordinates.

disc, where the actual distance between points is much less than the

Euclidean distance (since every point on the boundary is identified)—

notice how much Antarctica is stretched out in Figure 1.12. This gives

us our first example of a Riemannian metric (which for the time be-

ing we may simply think of as a notion of distance) on

D2,

apart from

the usual Euclidean one.

Exercise 1.7. Stereographic projections from the north and south

poles introduce two coordinate systems on the sphere minus the poles.

Find the coordinate transformation from one of those systems to the

other—that is, if a point on the sphere has coordinates (x, y) in the

coordinate system projected from the north pole and (x , y ) in the

projection from the south, find (x , y ) as a function of (x, y).

c. Parametric representations. While the idea of putting local

coordinates on a surface will turn out to be more useful in general, we

will occasionally have reason to deal with parametric representations.

There are two important distinctions between these two methods of

introducing coordinates on a surface.

First, local coordinates involve a map from the surface to a plane

domain, while a parametric representation is a map from a plane

domain to the surface. Formally, then, these two constructions are

mutual inverses.

The second distinction is that a local coordinate system usually

does not attempt to cover the entire surface by a single coordinate